The Ellipsoidal Shapes
of Nuclei

The neutrons and protons of nuclei are separately organized in shells. Remarkably however the maximum occupancies
of the shells are the same for neutrons and protons. Other than the shell structures the spatial organization of
neutrons and the protons has been a mystery.

Nuclei are believed to have near-spherical shapes; i.e., ellipsoidal. A quantity called the
electrical quadrupole moment (EQM) has been defined. The crucial matter is how can it be measured. There are
numerous methods that have been proposed for measuring the EQM of nuclei. While often these various methods give approximately
the same that is not always the case. N.J. Stone tabulated these measurements from individual studies published in the
journals of physics. They are available in his Atomic Data and Nuclear Data Tables which is updated every few years.

The expectation was that nuclei with filled shells would be spherical and thus have an electrical quadrupole
moment of zero. This expectation is examined in the table below. The table for the smaller nuclides includes only those measurements
obtained by the method labled NMR (nuclear magnetic resonance), including β-NMR (NMR with beta detection). A few cases in
which Stone re-evaluated the data were included.

The EQM has the dimensions of
electric charge times area. The area unit used for the tabulation is 10−24 cm², known as
a barn, as in "big as a barn door."

The Electric Quadrupole Moments of the Smaller Nuclides

Number of protons

Number ofNeutrons

Absolute Value of Electric QuadrupoleMoment (barns) |Q|

1

1

0.00286

3

3

0.0008

3

4

0.0406

3

5

0.0317

3

6

0.036

3

6

0.0253

3

8

0.031

4

5

0.0529

5

3

0.063

5

5

0.0847

5

6

0.0407

5

7

0.0134

5

8

0.037

5

9

0.0298

5

10

0.038

5

12

0.0386

6

5

0.032

7

5

0.0098

7

7

0.0208

7

9

0.018

7

11

0.0123

8

9

0.02578

8

10

0.036

8

11

0.0037

9

8

0.058

9

10

0.072

9

11

0.042

10

10

0.23

10

12

0.19

11

15

0.0053

11

16

0.0072

11

17

0.0395

11

18

0.086

The absolute values of the EQMs are given because in Stone's table in some cases the sign was not determined.

There is no clear pattern to the values and no apparent tendency for zero values at the magic numbers of nucleons.

The Difficulty of
Maintaining a Spherical
Structure for Nucleons

It is not possible to create a polyhedral structure of overall spherical shape that can maintain its structure by rotation
against attraction toward the center of the structure. If the rotation of the particles at or near the equator balances the force on them the ones
near the poles experience unbalanced force. What follows below is an explanation of how the dynamic appearance of
ellipsoidal shapes can be created in nuclei.

The Alpha Module
Structure of Nuclear Shells

The mass of a nucleus is less than the sum of the masses of the protons and neutrons that it is made of. The difference
is called its mass deficit. When the mass deficit is expressed in energy units via the Einstein equation E=mc²
it is called the binding energy of the nucleus. The binding energy has been computed for 2931 nuclides (types
of nuclei).

The incremental binding energy of a neutron in a nuclide with n neutrons and p protons is its binding energy less
that of a nuclide having one less neutron. The incremental binding energy of a neutron can be computed for
about 2820 (2931−111) nuclides. For all of the incremental binding energies of neutrons (IBEN) there is a sawtooth, odd-even
pattern in which the IBEN is higher for an even number of neutrons than for an odd number because of the formation
of a spin pair of neutrons. This means that whenever possible two neutrons form a spin pair. Here is an example of the
sawtooth pattern for IBEN.

Likewise the incremental binding energies of protons (IBEP) can be computed for
2769 (2931−162) nuclides. For all of these as well there is the sawtooth odd-even pattern indicating that that
proton-proton spin pairs are formed whenever possible. Here is an example of the sawtooth pattern for IBEP.

Whenever the number of protons is less than the number of neutrons the addition of another proton will result in the
formation of a neutron-proton spin pair. When the number of protons exceeds the number of neutrons no neutron-proton spin pair
is formed and so the level of the IBEP drops. The same thing happens to the IBEN when the number of neutrons passes from
a level below the number of protons to a level above it. The following graph shows the drop in incremental binding
energies when the number of one type of nucleon exceeds the number of the other type.

Thus whenever possible a proton and neutron form a neutron-proton spin pair. The fact that the level of incremental
binding energy drops when there is no excess of the other nucleon means that neutron-proton spin pair formation is exclusive;
i.e., a proton can form a neutron-proton spin pair with only one neutron, and likewise for a neutron.

This means that neutrons and protons are linked into chains.
A neutron is linked to one other
neutron. That neutron is linked through a spin pairing to a proton, which in turn is linked to another proton. Thus there
has to be a chain comprised of modules such as -n-p-p-n-, or equivalently -p-n-n-p-. These can appropriately be called
alpha modules. An alpha particle is just an alpha module in which the end nucleons link up.
More generally
there are several
alpha modules contained in a ring. Here is a depiction of a ring of four alpha modules. It is not intended to
depict realistically the arrangement of the nucleons; instead it is a symbolic representation.

These chains of alpha modules must close. Otherwise there would be a nucleon at one end of the chain or the other
without a linkage. An odd nucleon of
one type is left out of the chain

In addition to the binding energies reflecting the energies involved in the formation of substructures like spin pairs
there is the energies involved in the interaction of nucleons through the nuclear strong force. As noted previously
neutrons and protons are organized in shells. When one shell is filled any additional nucleon goes into the next shell.
The higher shells are at a greater distance from the nucleons in the other shells and therefore the interaction energies
are less. Therefore the incremental binding energies decrease for higher level shells.

The Dynamics of a
Ring of Alpha Modules

The neutrons and the protons in the same shell numbers are attracted to each other through the nuclear strong force.
In order to maintain stability these nucleons must rotate about their centers of mass. This means that the neutrons
and protons in a shell must rotate like a vortex ring; i.e., a so-called smoke ring.

A substructure linked together such as in a ring can be subject to motions that cannot occur for a single particle.
For example, a ring can rotate about a diameter line. For a circular ring this produces the dynamic appearance
of a sphere. There can be rotation about more than one diameter line. This reenforces the appearance of the
rotating ring as a sphere. A ring can also rotate about an axis through its center perpendicular to its plane. This
gives alone a trajectory for a single particle that is a toroidal helix. When this motion is combined with the
flipping motion of rotation about a diameter the trajectory of all the particles becomes very complicated and it
is easy to believe that each particle more or less covers a spherical shell.

These alpha module rings rotate in four modes. They rotate as a vortex ring to keep the neutrons and protons
(which are attracted to each other) separate. The vortex ring rotates like a wheel about an axis through its
center and perpendicular to its plane. The vortex ring also rotates like a flipped coin about two different diameters
perpendicular to each other.

The above animation shows the different modes of rotation occurring sequentially but physically
they occur simultaneously. (The pattern on the torus ring is just to allow the wheel-like rotation to be observed.)

Aage Bohr and Dan Mottleson found that the angular momentum of a nucleus (moment
of inertia times the rate of rotation) is quantized to h(I(I+1))½, where
h is Planck's constant divided by 2π and I is a positive integer. Using this result
the rates of rotation are found to be many
billions of times per second. Because of the complexity of the four modes of rotation each nucleon
is effectively smeared throughout a spherical shell. So, although the static structure of a nuclear shell is that
of a ring, its dynamic structure is that of a spherical shell.

At rates of rotation of billions of times per second
all that can ever be observed concerning the structure of nuclei is their dynamic appearances. This accounts for all
the empirical evidence concerning the shape of nuclei being spherical or near-spherical.

When there is an odd nucleon the appearance would be a sphere with a string or ribbon wrapped around it. This would
create a nonzero value for the quadrupole moment.